One-Way Functions Imply Secure Computation in a Quantum World

Abstract

Can a sender non-interactively transmit one of two strings to a receiver without knowing which string was received? Does there exist minimally-interactive secure multiparty computation that only makes (black-box) use of symmetric-key primitives? We provide affirmative answers to these questions in a model where parties have access to shared EPR pairs, thus demonstrating the cryptographic power of this resource.• First, we construct a one-shot (i.e., single message) string oblivious transfer (OT) protocol with random receiver bit in the shared EPR pairs model, assuming the (sub-exponential) hardness of LWE.Building on this, we show that secure teleportation through quantum channels is possible. Specifically, given the description of any quantum operation Q, a sender with (quantum) input ρ can send a single classical message that securely transmits Q(ρ) to a receiver. That is, we realize an ideal quantum channel that takes input ρ from the sender and provably delivers Q(ρ) to the receiver without revealing any other information. This immediately gives a number of applications in the shared EPR pairs model: (1) noninteractive secure computation of unidirectional classical randomized functionalities, (2) NIZK for QMA from standard (sub-exponential) hardness assumptions, and (3) a noninteractive zero-knowledge state synthesis protocol.

Cryptography from Pseudorandom Quantum States

“…In the classical setting, this notion of commitment schemes can be constructed from any length-tripling PRG [Nao91]. Recently, two independent works [GLSV21,BCKM21b] showed that commitment schemes with aforementioned properties imply maliciously secure multiparty computation protocols with quantum communication in the dishonest majority setting. Of particular interest is the work of [BCKM21b] who show that the transformation is robust even if the underlying commitment scheme has quantum communication.…”

“…In contrast, key-exchange is known to require computational assumptions if one can only use classical communication. More recently, the works of Grilo, Lin, Song and Vaikuntanathan [GLSV21] and Bartusek, Coladangelo, Khurana, and Ma [BCKM21b] demonstrate that quantum protocols for secure multiparty computation can be constructed from post-quantum one-way functions. On the other hand classical protocols for secure computation cannot be based (in a black-box way) on one-way functions alone [IR89].…”

Quantum commitments and signatures without one-way functions

“…[37] to the construction of Refs. [11,38]. If this is true, multi-party computation can also exist even if BQP = QMA.…”

“…For example, if quantum states are transmitted, statistically-secure key distribution is possible [9], although it is impossible classically. Furthermore, oblivious transfer [10] is possible with only (quantum-secure) one-way functions when quantum states are transmitted [11][12][13][14][15][16][17]. Classically, it is known to be impossible to construct oblivious transfer from only one-way functions [18,19].…”

“…It could be the case that in the quantum world commitments can be constructed from an assumption weaker than the existence of one-way functions. This possibility is mentioned in previous works [11,12], but no construction is provided.…”